Theory of branching and annihilating random walks

A systematic theory for the diffusion–limited reaction processes A + A → ⊘ and A → (m + 1)A is developed. Fluctuations are taken into account via the field–theoretic dynamical renormalization group. For m even the mean field rate equation, which predicts only an active phase, remains qualitatively correct near d_c = 2 dimensions; but below d′_c ≈ 4/3 a nontrivial transition to an inactive phase governed by power law behavior appears. For m odd there is a dynamic phase transition for any d ≤ 2 which is described by the directed percolation universality class.